The actions (REF ) on the bosonic fields can be cast into a cochain complexIt is explained in [1]} how this complex arises as the horizontal part of a double-complex which arises naturally for \(\mathcal {N}=2\)  localization as sketched above.
\(\begin{tikzcd}[column sep=scriptsize]0[r] & \Omega ^1_X(\mathfrak {g}_P)\oplus \Omega ^0(\mathfrak {g}_P)[r,"\eth "] & P^+_\omega \Omega ^2_X(\mathfrak {g}_P)\oplus \Omega ^0_X(\mathfrak {g}_P)\oplus \Omega ^0_X(\mathfrak {g}_P)[r] & 0\end{tikzcd},\) 
